# NBHM (National Board for Higher Mathematics) MSc and MA Mathematics: Questions 47 - 54 of 101

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## Question 47

Appeared in Year: 2009

### Write in Short

Let

Write down the residues of at each of its poles

## Question 48

Appeared in Year: 2007

### Describe in Detail

Essay▾

Let C denote the Boundary of the square whose sides are given by the lines . Assume that C is described in the position sense, i.e.. anticlockwise. Evaluate

### Explanation

To evaluate

Take

Here is a pole of order 1 and lies within the square.

These are simple poles which lie outside the square

So they will not contribute towards the residue of

By Cauchy residue theorem

## Question 49

Appeared in Year: 2014

### Describe in Detail

Essay▾

Let be differentiable at and let . Evaluate

### Explanation

Now

So this is form (Exponential form)

Now using the result

Or if

Then

… eq. (1)

form

From (1)

… (1 more words) …

## Question 50

Appeared in Year: 2009

### Describe in Detail

Essay▾

Let C be the contour consisting of the lines , described counterclockwise in the plane. Compute

### Explanation

is the contour shown in the figure

To evaluate

Now is a pole of order 1 (simple pole)

Also lies within C

=

By Cauchy residue theorem

## Question 51

Appeared in Year: 2017

### Describe in Detail

Essay▾

Let Write down the Taylor Expansion of f in the neighbourhood of .

### Explanation

Here

Put

This is the required Taylor expansion of f in the neighbourhood of

## Question 52

Appeared in Year: 2015

### Describe in Detail

Essay▾

In each of the following cases, state whether the given series is absolutely convergent, conditionally convergent or divergent

a)

b)

c)

### Explanation

Option (a) The given series is

Where

Now comparing it with

1)

2)

3)

By Leibnitz test (alternating series test)

The given series is convergent

Which is finite & non – zero

converge or diverge together

But is divergent by p – test

is divergent (By comparison test)

is not absolutely convergent

is conditionally convergent

Option (b) The given series i…

… (44 more words) …

## Question 53

Appeared in Year: 2006

### Question

MCQ▾

Pick out series which are absolutely convergent.

### Choices

Choice (4)Response

a.

b.

c.

d.

Both a. and b. are correct

## Question 54

Appeared in Year: 2005

### Describe in Detail

Essay▾

What are the value of for which the following series is convergent?

### Explanation

Let the omen series be denoted by … eq. (1)

Where

Since for absolute convergence, we take

Now is convergent if … eq. (2)

And divergent if (By p-test)

Also eq. (1) is convergent for by Leibnitz test

is convergent by Leibnitz test for … eq. (3)

From eq. (2) & (3)

We get eq. (1) is convergent

… (3 more words) …